
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm}   % This is your set screw

%%文档的题目、作者与日期
\author{王立庆（2022级数学与应用数学1班）}
%\author{LQW }
\title{高等代数(一)复习1-13}
%\date{\vspace{-3ex}}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
\date{2022 年 12 月 27 日}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %1
已知行列式 $D=\begin{vmatrix} 1&2&3&4 \\ 5&1&2&3 \\ 4&5&1&2 \\ 3&4&5&1 \end{vmatrix}$，
求代数余子式 $A_{23}$ 的值。
 
\begin{enumerate}
\item $3$.
\item $-3$.
\item $5$.
\item $-5$.
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(d). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %2
七阶行列式 $D=|a_{ij}|_{7\times 7}$ 写出来一共有多少项？
其中一项 $a_{31}a_{27}a_{52}a_{14}a_{73}a_{46}a_{65}$ 前面的正负号是什么？
\begin{enumerate}
\item 720 项，正号。
\item 720项，负号。
\item 5040项，正号。
\item 5040项，负号。
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(c). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %3
设 $n$ 阶行列式  
%\begin{eqnarray*}
$D_n=\begin{vmatrix} 
2&1&0&\cdots &0&0 \\ 
1&2&1&\cdots &0&0 \\ 
0&1&2&\cdots &0&0 \\  
\vdots&\vdots&\vdots& &\vdots&\vdots  \\ 
0&0&0&\cdots &2&1 \\
0&0&0&\cdots &1&2   
\end{vmatrix}.
$%\end{eqnarray*}
则下述递推式正确的是哪个？
\begin{enumerate}
\item $D_n = 2D_{n-1} - D_{n-2}$. 
\item $D_n = 2D_{n-1} + D_{n-2}$. 
\item $D_n = D_{n-1} - 2D_{n-2}$. 
\item $D_n = D_{n-1} + 2D_{n-2}$. 
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(a). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %4
将下述矩阵 $A$ 用行初等变换化为行最简形 $B$. 求矩阵 $B$ 的所有元素的和。
\begin{eqnarray*}
A=\begin{bmatrix} 1&2&1&3 \\ 1&3&1&4 \\ 2&2&0&4  \end{bmatrix}. 
\end{eqnarray*}

\begin{enumerate}
\item 4.
\item 5.
\item 6.
\item 7.
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(b). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %5
求出线性方程组的系数矩阵和增广矩阵的秩，判断该线性方程组是否有解。
\begin{eqnarray*}
\left\{\begin{array}{rrrrrr}
9x_1 & +6x_2 & +7x_3 & + 10x_4 &=& 3, \\
6x_1 & +4x_2 & +2x_3 & + 3x_4 &=& 2, \\
3x_1 & +2x_2 & -11x_3 & -15x_4 &=& 1. 
\end{array}\right. 
\end{eqnarray*}

\begin{enumerate}
\item 因为 $R(A)=2, R(\bar{A})=2$, 所以有解。
\item 因为 $R(A)=2, R(\bar{A})=2$, 所以无解。
\item 因为 $R(A)=2, R(\bar{A})=3$, 所以有解。
\item 因为 $R(A)=2, R(\bar{A})=3$, 所以无解。
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(a). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %6
设矩阵 $A=\begin{bmatrix} 1&2 \\ 3&0  \end{bmatrix}$. 求矩阵 $B=AA^t+A^2+2A$ 的所有元素的和。

\begin{enumerate}
\item 48.
\item 49.
\item 50.
\item 51.
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(c). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %7
通过第三类行和列的初等变换，实现矩阵的下述初等变换，
\begin{eqnarray*}
A=\begin{pmatrix}1&2&1 \\ 1&3&2 \\ 2&3&4 \end{pmatrix}
\to%\xrightarrow[\text{ }]{\text{一系列的第三类初等变换 }}
\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&d \end{pmatrix}=B. 
\end{eqnarray*}
则 $d$ 的值是多少？

\begin{enumerate}
\item 1.
\item 2.
\item 3.
\item 4.
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(c). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %8
求解矩阵方程
$\begin{pmatrix}1&2 \\ 3&7 \end{pmatrix}X = \begin{pmatrix}5&6\\ 7&11 \end{pmatrix}$. 
则矩阵 $X$ 的所有元素的和是多少？

\begin{enumerate}
\item 24.
\item 25.
\item 26.
\item 27.
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(c). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %9
行列式乘积公式是说，对任意 $n$ 阶矩阵 $A$ 与 $B$ 都有 $\det(AB)=\det(A)\det(B)$. 
关于这个定理的证明，下述说法中，不正确的是哪个？

\begin{enumerate}
\item 当 $n$ 等于1或2时，可以直接计算等式两边。
\item 当 $A$ 是初等矩阵时，根据左乘初等矩阵相当于行初等变换，以及行列式的性质， 可以得证。
\item 当 $A$ 是可逆矩阵时，将$A$ 写成一些初等矩阵的乘积，可以得证。
\item 当 $A$ 是不可逆矩阵时，矩阵 $A$ 的行最简形有一列全为零，可证等式左边也是零。
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(d). 


}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %10
用行初等变换从 $AX=B$ 求得 $X=A^{-1}B$ 的原理包括下述几步。
其中不正确的是哪个？
\begin{enumerate}
\item  对矩阵 $A$ 做行初等变换等价于在矩阵 $A$ 的左边乘以相应的初等矩阵。
\item  由 $AX=B$ 可得 $P(AX)=PB$, 再由乘法的交换律，可得 $(PA)X=PB$. 
\item  选取一些初等矩阵 $P_1,P_2,\cdots, P_s$ 使得 $P_s\cdots P_2P_1A=E$. 
\item  可得 $X=P_s\cdots P_2P_1B$. 这正好是对矩阵 $B$ 进行一系列相同的行初等变换。
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{red}解答：(b). 


}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
